A real estate agent wants to predict the selling price of single-family homes from the size of each house. A scatterplot created from a sample of houses shows an exponential relationship between price, in thousands of dollars, and size, in 100 square feet. To create a linear model, the natural logarithm of price was taken, and the least-squares regression line was given as:

[tex]
\ln (\widehat{\text{price}}) = 2.08 + 0.11 (\text{size})
[/tex]

Based on the model, which of the following is closest to the predicted selling price for a house with a size of 3,200 square feet?

A. [tex] \$54,500 [/tex]

B. [tex] \$270,000 [/tex]

C. [tex] \$354,000 [/tex]

First, substitute the size of the house (3,200 square feet) into the equation. Remember that size is in units of 100 square feet, so divide 3,200 by 100 to convert:

[tex]\ln (\widehat{\text{price}}) = 2.08 + 0.11 \times 32[/tex]

[tex]\ln (\widehat{\text{price}}) = 2.08 + 3.52[/tex]

[tex]\ln (\widehat{\text{price}}) = 5.60[/tex]

Now, solve for the predicted price by taking the exponential of both sides to undo the natural logarithm:

[tex]\widehat{\text{price}} = e^{5.60}[/tex]

Using a calculator, [tex]e^{5.60}[/tex] is approximately 270.43. Since the price is in thousands of dollars, the predicted selling price is approximately $270,430.